Question

What is the next item in the sequence $1,4,16,64,256?$

Answer 1

Hint: Observe the above sequence first find the ratio of the terms in consecutive and will check the pattern for the sequence whether the given sequence is geometric progression or not.

Complete step-by-step solutions:
In the geometric progression is the sequence in which the succeeding element is obtained by multiplying the preceding number by the constant and the same continues for the series. The ratio between the two remains the same.
Now,
Ratio between the two consecutive terms is –
$\dfrac{{256}}{{64}} = \dfrac{{64}}{{16}} = \dfrac{{16}}{4} = \dfrac{4}{1} = 4$
Therefore, the given sequence is $4$
And hence, the next term will be $256 \times 4 = 1024$

Thus the required answer is 1024.

Note: Know the difference between the arithmetic and geometric progression and apply the concepts accordingly. In arithmetic progression, the difference between the numbers is constant in the series whereas, the geometric progression is the sequence in which the succeeding element is obtained by multiplying the preceding number by the constant and the same continues for the series. The ratio between the two remains the same.
The geometric Progression is often called GP and arithmetic progression is called AP. General form of geometric progression is –
${\text{Sum = a + ar + a}}{{\text{r}}^2} + a{r^3} + .....{\text{ + a}}{{\text{r}}^{n - 1}}$